Mixed Jacobi-like forms of several variables

Jacobi-like forms of one variable are formal power series with holomorphic coefficients satisfying a certain transformation formula with respect to the action of a discrete subgroup Γ of SL(2,R), and they are related to modular forms for Γ, which of course play a major role in number theory. Indeed, by using this transformation formula, it can be shown that that there is a one-to-one correspondence between Jacobi-like forms whose coefficients are holomorphic functions on the Poincaré upper half-plane and certain sequences of modular forms of various weights (cf. [1, 12]). More precisely, each coefficient of such a Jacobi-like form can be expressed in terms of derivatives of a finite number of modular forms in the corresponding sequence. Jacobi-like forms are also closely linked to pseudodifferential operators, which are formal Laurent series for the formal inverse ∂−1 of the differentiation operator ∂ with respect to the given variable (see, e.g., [1]). In addition to their natural connections with number theory and pseudodifferential operators, Jacobi-like forms have also been found to be related to conformal field theory in mathematical physics in recent years (see [2, 10]). The generalization of Jacobi-like forms to the case of several variables was studied in [8] in connection with Hilbert modular forms, which are essentially modular forms of several variables. As it is expected, Jacobi-like forms of several variables correspond to sequences of Hilbert modular forms. Another type of generalization can be provided by considering mixed Jacobi-like forms of one variable for a discrete subgroup Γ⊂ SL(2,R), which are associated to a holomorphic map of the Poincaré upper half-plane that is equivariant with respect to a homomorphism of Γ into SL(2,R) (cf. [7, 9]). Mixed Jacobi-like